Abstract: In this paper we study connected regular linear algebraic monoids. If is a representation of a reductive group , then the Zariski closure of in is a connected regular linear algebraic monoid with zero. In we study abstract semigroup theoretic properties of a connected regular linear algebraic monoid with zero. We show that the principal right ideals form a relatively complemented lattice, that the idempotents satisfy a certain connectedness condition, and that these monoids are -regular. In we show that when the ideals are linearly ordered, the group of units is nearly simple of type . In , conjugacy classes are studied by first reducing the problem to nilpotent elements. It is shown that the number of conjugacy classes of minimal nilpotent elements is always finite.